3.5 Location Estimation in Unknown Path-Loss Model
3.6.2 Square area with ANs around the edges with incorrect PLE
In the second scenario, a 2-D area with dimensions 50 m × 50 m is considered with 8 ANs around the edge at [0, 0], [50, 0], [50, 50], [0, 50], [25, 0], [50, 25], [0, 25], [25, 50]. Again instead of assuming a single TN, 20 TNs are assumed within the network to observe the average performance. This is illustrated in Fig. 3.6. For the LJE-ref, LLS-ref implementation, [25, 50] is taken as the reference AN. The simulation is executed η = 300 times independently. Equal noise variance for all ˆ
zi i.e. σi2 = σ2 is assumed and the actual value of PLE α0 = 3. The incremental
step to estimate ˆα in (3.31) is 0.1.
3.6.2.1 Bias of ˆα
For the network described above, simulations to estimate α are done for 20 TNs. Table 3.1 shows the average of the obtained ˆα values for various levels of noise in
3.6 Simulation Results
the observed ˆzi. It is evident that although the estimated values are close to the
actual PLE α0, yet there is a positive bias in the estimates. This bias effects the
estimation of the location coordinates and is described in subsection 3.6.2.3.
3.6.2.2 Error analysis
In Fig. 3.7, simulation and error analysis results are shown, error in terms of the RMSE subject to incorrect PLE assumption and noise is given. The error analysis accurately predicts the performance of the estimator (LLS-ref). It is evident that inaccurate PLE assumption can result in substantial error in location estimates. In Fig. 3.8, the bias due to incorrect PLE and noise is shown. It is also observed that comparatively degraded performance in terms of both RMSE and the bias is observed for ˇα < α0 than for ˇα > α0. These results confirm the error analysis
results obtained in chapter 2 for the ML technique.
3.6.2.3 Performance comparison of LJE and LLS
Fig. 3.9 shows the performance comparison between the variants of LLS and LJE. The RMSE of all estimators is compared while increasing the variance in the path-loss noise. It is observed that there is no considerable performance difference between the three different approaches of LLS. It is also observed that performance of LJE is close to LLS with LJE-avg and LJE-comb performing slightly better than LJE-ref. However, it is also observed that at certain points due to the bias of the LJE in estimating ˆα, its performance exceeds that of
the LLS. This is a counter intuitive phenomenon but is inherent with biased estimators. There is considerable gap between the performance of the estimators and the CRB, the reason for this is discussed in the Chapter 4.
3.6 Simulation Results −0.60 −0.4 −0.2 0 0.2 0.4 0.6 10 20 30 40 50 60 ∆α Av er age R M S E (m ) Simulation σ2= 6 Simulation σ2= 3 Error analysis σ2= 6 Error analysis σ2= 3 Error analysis σ2= 0
Figure 3.7: Error analysis: RMSE and simulation for incorrect PLE.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 −4 −2 0 2 4 6 8 10 12 14 16 ∆α Av er age B ias (m ) Simulation σ2= 6 Simulation σ2= 3 Error analysis σ2= 6 Error analysis σ2= 3 Error analysis σ2= 0
3.6 Simulation Results 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 σ2 Av er age R M S E (m ) LLS-ref LLS-avg LLS-comb LJE-ref LJE-avg LJE-comb CRB CRB-α
Figure 3.9: Performance comparison between variants of LLS and LJE, the CRB
(with known α) and the CRB-α (estimated α).
3.6.2.4 LJE as initial estimate for LCJE
In the simulation shown in the Fig. 3.10, the performance of the LCJE is com- pared when given a random value as the initial estimate θ1 and when the estim- ated value of LJE, ˆθLJ E, is selected as θ1. It is seen that there is considerable
performance improvement when θ1 = ˆθLJ E. Indeed the performance of LCJE
with a random θ1 can be improved by taking more iterations. Nevertheless, it is seen that only with k = 3 iterations, for θ1 = ˆθLJ E reach near optimal per-
formance. On the other hand, for an arbitrarily chosen initial estimate, there is degraded performance even for k = 6 iterations. Thus the advantages of this approach are twofold; i) there is obvious performance improvement in terms of power consumption and computational time taken with smaller number of it- erations. ii) the requirement for the selection of a close initial point to avoid convergence to local minima is bypassed.
3.7 Summary 0 1 2 3 4 5 6 0 1 2 3 4 5 σ2 Average RMSE (m) Arbitrary θ1, k = 3 Arbitrary θ1, k = 5 Arbitrary θ1, k = 6 θ1= ˆθ LJE, k = 3 CRB-α
Figure 3.10: Performance of LCJE for arbitrary θ1and θ1 = ˆθLJ E.
3.7 Summary
In this chapter, a simplified low complexity RSS based location estimator for unknown path-loss model is proposed. The error analysis for incorrect PLE as- sumption was done. Based on the linear model, analytical expressions for the RMSE and bias were derived. It was seen via simulation that analysis results accurately predicts the performance of the linear estimator. For correct PLE assumption, the performance of the estimator is unbiased if the TN is at equal distance from all ANs. It was also observed that use of an incorrect PLE has dramatic impact on the accuracy of location estimates. Both the MSE and bias are large for ˇα < α0 than for ˇα > α0. Next, a simplistic technique to estimate
the PLE by optimizing a single variable function was devised. Simulation res- ults show that this technique has acceptable performance though the estimates are biased. In order to achieve even better accuracy, the LJE results are used as the initial estimate for more computationally intense but optimal algorithm
3.7 Summary
and showed via simulation that this performs considerably better with a smaller number of iterations in comparison with an arbitrary initial estimate. For future work, the joint estimation of different PLE for each link and the location will be investigated. Furthermore, in such scenario the geometry of ANs and its impact on location accuracy will be studied.
4 Optimising Linear Least Squares
Solution to RSS Localisation
The material in this chapter has been published in the paper:
• N. Salman, M. Ghogho, and A. H. Kemp, “Optimized Low Complexity Sensor Node Positioning in Wireless Sensor Networks,” EEE Sensors Jour-
nal.,vol.14, no.1, pp.39,46, Jan. 2014.
4.1 Overview
It was shown in chapter 2 that due to the non-linear nature of the localisation problem, location estimation via RSS (and also for ToA) can be achieved us- ing maximum likelihood (ML) techniques that commonly operate in an iterative fashion. Generally, a close initial estimate of location is required for the ML algorithm. Furthermore, the ML technique due to its iterative nature is high in complexity. On the other hand, location can also be estimated by employing a low complexity linear least squares (LLS) approach as discussed in chapter 3. The LLS technique does not require a close initial estimate and is of low com- plexity as it does not require multiple iterations. However it was noticed that the LLS technique performs sub-optimally. Hence the LLS technique needs further
4.1 Overview
optimization to achieve acceptable results. In addition it was also noted from 3.9 that the conventional CRB does not tightly bound the performance of the LLS. Hence a new bound needs to be derived.
In this chapter the performance of the LLS RSS location estimator is analysed and improvement is proposed. The linear model in this chapter is modified to account for different PLEs and noise variance for each link. The basic concept behind the LLS technique is that instead of using individual readings from ANs, readings from AN pairs are first formulated (subtracted from each other) to linearise the non-linear system of equations. Generally, a reference node has to be chosen and paired with all other ANs. However, random selection of an AN as a reference can cause performance degradation. Other techniques to linearise the system include averaging the readings from all ANs and then pairing them with individual AN. Finally, pairing each AN with every other AN can be used for linearisation. The system performance can be optimized by choosing an optimal reference AN and pairing it with all other ANs. In this chapter, a technique for optimal reference AN selection using the RSS signals is devised. In order to further improve the performance, the correlation between the (now linear) RSS readings is used and a weighted least squares (WLS) algorithm is proposed. For optimized performance the optimal AN selection for the WLS method is also given in the chapter. In order to compare the MSEs of estimators, the Cramer-Rao bound has been extensively used as a benchmark. For ML algorithms, the CRB on location estimated has been derived for ToA in [45, 46] and for RSS systems in [18]. However, since the LLS method is not based on individual readings, the CRB given in [18] does not tightly bound the performance of the LLS-RSS estimator. For ToA LLS technique the CRB is given in [44]. The ToA linear CRB in [44] does not lower bound the performance of the RSS system due to different signal and noise model. In this chapter, the linear CRB is derived to tightly bound the
4.2 System Model
performance of the LLS and WLS algorithm based on RSS range estimation. To sum up, the main contributions of this chapter are as follows:
• WLS algorithm for the linear model is proposed.
• Optimal anchor selection for both LLS and WLS methods is proposed. • Linear CRB for RSS systems is derived.
Simulation results show that the linear CRB is significantly larger than the exact CRB and is thus more realistic in lower bounding the performance of RSS systems using the linear model. It is shown via simulations that the performance of the LLS estimator improves considerably when the optimal reference AN is used. The system performance is further improved using the WLS algorithm with optimal AN selection.
The rest of the chapter is organized as follows. Section 4.2 presents the problem statement and the system model. In section 4.3, the modified linear RSS model and the LLS solution is presented. In section 4.4, the WLS algorithm is proposed. In section 4.5, the optimal reference AN selection technique is presented. In section 4.6, linear CRB is derived. Finally, in section 4.7, the simulation results are discussed which are followed by conclusions.
4.2 System Model
Unless otherwise specified, same notations as in chapter 2 are used in this chapter. The signal model (before linearisation) is similar to that in chapter 2, with a different PLE αi used for each anchor now. The signal model will be rewritten
here for easy understanding.
A two dimensional (2-D) network is considered, consisting of a TN which has unknown coordinates θ = [x, y]T (θ ∈R2) that are to be estimated, and M ANs
4.2 System Model
with known locations θi = [xi, yi]T(θi ∈R2) for i = 1, ..., M. The received power
at the ANs due to random shadowing is log-normally distributed. This model is based on empirical results obtained in [36, 37]. Thus the distance di between the
TN and the ith AN, is related to the path-loss at the ith AN, L
i, and the PLE,
αi, as [35]
Li =L0+ 10αilog10di+ wi, (4.1)
where L0 is the path-loss at the reference distance d0 (d0 < di, and is normally
taken as 1 m) and wi is a zero-mean Gaussian random variable with known
variance representing the log-normal shadowing effect, i.e. wi ∼ (N (0, σi2)). The
PLEs are assumed to be known via prior channel modelling or accurate estimation [32]. The path-loss is calculated as
Li = 10 log10Pt− 10 log10Pi (4.2)
where Pt is the transmit power at the TN and Pi is the received power at the ith
AN. The distance di is given by
di =
q
(x − xi)2+ (y − yi)2. (4.3)
The observed path-loss (in dB) from d0 to di, zi =Li−L0, can be expressed as
zi = fi(θ) + wi, i = 1, ..., M (4.4)
where fi(θ) = γαiln di and γ = ln 1010 . In a vector form,
z = f (θ) + w, (4.5)
where z = [z1, ..., zM] T